Ammari H., Benkirane A., Touzani A. (editors) Recent Developments in Nonlinear Analysis

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i.e.,

k|T

)|k

≤ c

. (16)

On the other hand, we have,

{|u

|>k}

|g(x, u

, ∇u

)| ≤

Ω

||T

)| dx ≤ kc

, (17)

which implies

{|u

|>k}

j=1

∂u

∂x

≤

{|u

|>k}

| dx. (18)

Then, by (13), (16) and (17) with k > ρ

, we obtain

k|u

i=1

{|u

>k|}

∂u

∂x

dx +

i=1

Ω

∂T

)

∂x

≤

{|u

|>k}

|g(x, u

, ∇u

)| + c

+ c

≤ c

where c

i = 1, 2, . . . , are various positive constants. Then

k|u

≤ c

where c is some positive constant. Hence, we can extract a subsequence still

denoted by u

such that,

* u in W

1,p

(Ω, w) and a.e. in Ω. (19)

Step (2) Convergence of the positive part of u

Our objective in this step is to prove that

→ u

strongly in W

1,p

(Ω, w). (20)

Assertion (i) We prove that:

lim

ε→0

Ω

[a(x, u

, ∇u

) −a(x, u

, ∇u

)]∇(u

− u

)

dx = 0. (21)

Let k be a positive constant greater than ρ

. We use v

= T

− u

)

as a test function in (P

), this is an admissible test function which due by

Lemmas 2.3 and 2.4. Multiplying (P

) by v

= T

− u

)

, we obtain

hAu

, v

i +

Ω

g(x, u

, ∇u

dx = hf

, v

If v

(x) > 0, we have u

(x) > 0 and from (11) g(x, u

, ∇u

) ≥ 0, then

hAu

, v

i ≤ hf

, v

i i.e.,

Ω

a(x, u

, ∇u

)∇v

dx ≤ hf

, v

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351

Since u

= u

in {x ∈ Ω, u

(x) > u

(x)}, then

Ω

a(x, u

, ∇u

)∇T

− u

) dx ≤ hf

, T

− u

)i,

which implies

lim

ε→0

Ω

[a(x, u

, ∇u

) −a(x, u

, ∇u

)]∇T

− u

)

dx = 0. (22)

On the other hand, since again that u

(x) = u

(x) in the set {x ∈

Ω, u

(x)−u

(x) > 0}, then by using (8) and Young’s inequality, we obtain

−u

[a(x, u

, ∇u

) −a(x, u

, ∇u

)]∇(u

− u

)

≤

[a(x, u

, ∇u

) − a(x, u

, ∇u

)]∇(u

− u

)

≤ c



(x) dx +

σ dx

i=1

∂u

∂x

dx +

i=1

∂u

∂x



thanks to (18), we deduce

−u

[a(x, u

, ∇u

) − a(x, u

, ∇u

)]∇(u

− u

)

≤ c



(x) dx +

σ dx +

|dx

i=1

∂u

∂x



(23)

If k tends to inﬁnity the right hand side of (23) converges to zero with

respect to ε.

Combining (22) and (23), we conclude (21).

Assertion (ii) We claim that

lim

ε→0

Ω

[a(x, u

, ∇u

) − a(x, u

, ∇u

)]∇(u

− u

)

−

dx = 0. (24)

Indeed, take z

= u

−T

), we shall use the test function v

= ϕ

−

)

with ϕ

(s) = se

λs

in (P

). If v

(x) 6= 0, we have 0 ≤ u

(x) ≤ k, hence

−

∈ L

∞

(Ω) and since z

−

∈ W

1,p

(Ω, w), hence by Lemma 2.2, we have

∈ W

1,p

(Ω, w). Multiplying (P

) by v

, we obtain

Ω

a(x, u

, ∇u

)∇z

−

) dx+

Ω

g(x, u

, ∇u

)ϕ

−

) dx = hf

, ϕ

−

)i.

(25)

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352

Firstly, we study the term of the right hand side, since ϕ

−

) 6= 0 where

0 ≤ u

(x) ≤ k, then ϕ

−

) is bounded in L

∞

(Ω), then by Lebesgue

dominated convergence theorem, we obtain

lim

ε→0

Ω

−

) dx =

Ω

fϕ

− T

))

−

) dx.

On the other hand, we study the term of left hand side as in the proof of

assertion (4.6) in.

Consequently, we conclude that

−lim sup

ε→0

Ω

[a(x, u

, ∇u

)−a(x, u

, ∇T

(u)

)]∇(u

−T

(u)

)

−

≤ 0. (26)

Finally, we deduce the assertion (24) as in the proof of (21) in.

Combining (21) and (24), we obtain

lim

ε→0

Ω

[a(x, u

, ∇u

) − a(x, u

, ∇u

)]∇(u

− u

) dx = 0. (27)

Which and using Lemma 3.1, we conclude (20).

Step(3) Convergence of the negative part of u

The proof of this step is similar to the step (2), it is suﬃcient to take

= T

−

−u

−

)

and v

= ϕ

−

−u

−

)

−

as test function in (P

). Then,

we can prove that

−

→ u

−

strongly in W

1,p

(Ω, w). (28)

Step(4) Strong convergence of u

and passing to the limit.

From (20) and (28), we conclude that for a subsequence,

→ u strongly in W

1,p

(Ω, w) and a.e. in Ω. (29)

Consequently, we deduce that

∇u

→ ∇u in Π

i=1

(Ω, w

) and a.e. in Ω.

Reasoning as in the proof of Theorem 3.2 in,

we can prove that

g(x, u

, ∇u

) → g(x, u, ∇u) strongly in L

(Ω). (30)

Finally, thanks to (29) and (30), it is easy to pass to the limit in

Ω

a(x, u

, ∇u

)∇v dx +

Ω

g(x, u

, ∇u

)v dx =

Ω

v dx

for all v ∈ W

1,p

(Ω, w) ∩ L

∞

(Ω).

Then, we obtain

Ω

a(x, u, ∇u)∇v dx +

Ω

g(x, u, ∇u)v dx =

Ω

fv dx

for all v ∈ W

1,p

(Ω, w) ∩ L

∞

(Ω).

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353

3.1. Example

Some ideas of this example come from.

Let Ω be a bounded domain

of IR

(N ≥ 1), satisfying the cone condition. Let us consider the

Carath´eodory functions:

(x, s, ξ) = w

|ξ

p−1

sgn(ξ

) for i = 1, ..., N

g(x, s, ξ) = ρs|s|

i=1

|ξ

, ρ > 0, r > 0

where w

(x) (i = 0, 1, ..., N ) are a given weight functions strictly positive

almost everywhere in Ω. We shall assume that the weight functions satisfy,

(x) = w(x), x ∈ Ω, for all i = 0, ..., N. Then, we can consider the

Hardy inequality (6) in the form,



Ω

|u(x)|

σ(x) dx



≤ c



Ω

|∇u(x)|



It is easy to show that the a

(x, s, ξ) are Carath´eodory functions satisfying

the growth condition (8) and the coercivity (10). Also the Carath´eodory

function g(x, s, ξ) satisﬁes the conditions (11), (12) and (13) with |s| ≥

= 1 and ρ

= ρ > 0. On the other hand, the monotonicity condition is

veriﬁed, in fact,

i=1



(x, s, ξ) − a

(x, s,

ξ)



−



= w(x)

i=1



|ξ

p−1

sgnξ

− |

p−1

sgn



−



> 0

for almost all x ∈ Ω and for all ξ,

ξ ∈ IR

with ξ 6=

ξ, since w > 0 a.e. in Ω.

In particular, let us use the special weight functions w and σ expressed in

terms of the distance to the boundary ∂Ω. Denote d(x) = dist(x, ∂Ω) and

set

w(x) = d

(x), σ(x) = d

(x).

In this case, the Hardy inequality reads



Ω

|u(x)|

(x) dx



≤ c



Ω

|∇u(x)|

(x) dx



The corresponding imbedding is compact if:

i) For, 1 < p ≤ q < ∞,

λ < p − 1,

−

+ 1 ≥ 0,

−

+ 1 > 0. (31)

December 28, 2009 12:15 WSPC - Proceedings Trim Size: 9in x 6in recent

354

ii) For, 1 ≤ q < p < ∞,

λ < p − 1,

−

+ 1 > 0. (32)

Remark 3.4. Condition (31) or (32) are suﬃcient for the compact imbed-

ding (7) to hold (see for example

Example 1,

Example 1.5, p.34 and

theorem 19.17 and 19.22).

Finally, the hypotheses of theorem 3.1 are satisﬁed, therefore the problem

(P) has at least one solution.

References

1. Y. Akdim, E. Azroul and A. Benkirane, Existence of Solution for

Quasilinear Degenerated Elliptic Equations, Electronic J. Diﬀ. Equ.,

Vol. 2001 N 71, (2001) pp 1-19.

2. Y. Akdim, E. Azroul and A. Benkirane, Existence of Solution

for Quasilinear Degenerated Elliptic Unilateral Problems, Annale

Math´ematique Blaise Pascal Vol. 10 (2003) pp 1-20.

3. Y. Akdim, E. Azroul and A. Benkirane, Existence results for Quasi-

linear Degenerated equation via strong convergence of trancations, Re-

vista Matematica Complutense , Vol.17 (2004), pp 359-379..

4. A. Bensoussan, L. Boccardo and F. Murat, On a non linear par-

tial diﬀerential equation having natural growth terms and unbounded

solution, Ann. Inst. Henri Poincar´e 5 N 4 (1988), 347-364.

5. L. Boccardo and T. Gallou

et, Strongly MonlinearElliptic Equa-

tions Having Natural Growth Terms and L

Data,Nonlinear Analysis

Theory Methods and Applications, Vol 19, N 6 (1992) 573-579.

6. L. Boccardo, T. Gallou

et and F. Murat, A uniﬁed presentation

of two existence results for problems with natural growth,in Progress in

PDE, the Metz surveys 2, M. Chipot editor, Research in Mathematics,

Longman, 296 (1993), 127-137.

7. P. Drabek, A. Kufner and V. Mustonen, Pseudo-monotonicity and

degenerated or singular elliptic operators, Bull. Austral. Math. Soc.

Vol. 58 (1998), 213-221.

8. P. Drabek, A. Kufner and F. Nicolosi, Non linear elliptic equations,

singular and degenerate cases, University of West Bohemia, (1996).

9. P. Drabek and F. Nicolosi, Existence of Bounded Solutions for Some

Degenerated Quasilinear Elliptic Equations, Annali di Mathematica

pura ed applicata (IV), Vol. CLXV (1993), pp. 217-238.

10. B. Opic and A. Kufner, Hardy-type inequalities, Pitman Research

Notes in Mathematics Series 219(Longman Scientiﬁc and Technical,

Harlow, 1990).

December 28, 2009 12:15 WSPC - Proceedings Trim Size: 9in x 6in recent

355

Combined fuzzy sliding mode controller for a class of nonlinear

systems

Y. Alaoui Haﬁdi

Laboratoire d’Electronique Signaux Syst`emes et Informatique (L.E.S.S.I)

D´epartement de physique, Facult´e des Sciences Dhar Mehraz

B.P: 1796, 30000 Fes-Atlas, Morocco

J. Boumhidi

Laboratoire d’Informatique, Math´ematiques, Automatique et d’Opto´electronique

(L.I.M.A.O)

Facult´e Polydisciplinaire de Taza

Route d’Oujda, B.P: 1223 Taza Morocco

E-mail: jboumhidi@yahoo.fr

I. Boumhidi

Laboratoire d’Electronique Signaux Syst`emes et Informatique (L.E.S.S.I)

D´epartement de physique, Facult´e des Sciences Dhar Mehraz

B.P: 1796, 30000 Fes-Atlas, Morocco

In this study, we present an adaptive combined fuzzy sliding mode controller

for a class of nonlinear systems with unknown nonlinear dynamics. The pro-

posed control strategy is based on the fuzzy logic and sliding mode control

technique (SMC). The adaptive fuzzy logic system type Takagi-Sugeno (T-S)

is used to approximate the unknown dynamics of system. In order to assure

the robustness on stability and tracking performance, the sliding mode control

term is added in the control law. The latter consists to suppress the inﬂuence of

external disturbances and attenuate fuzzy approximation error. Simulation re-

sults illustrate the good performances of the system when the proposed control

technique is applied.

Keywords: Fuzzy logic; Sliding mode control; Takagi-Sugeno-type system.

1. Introduction

Fuzzy logic, as one of the most useful approaches for utilizing expert knowl-

edge, has been an active ﬁled of research during the past decade,

Fuzzy

logic control has found promising applications for a wide based on the uni-

versal approximation theorem,

stable variety of systems speciﬁcally ap-

December 28, 2009 12:15 WSPC - Proceedings Trim Size: 9in x 6in recent

356

plicable to plants that are mathematically poorly modeled.

Based on the

universal approximation capability, many eﬀective adaptive fuzzy control

schemes have been developed to incorporate with human expert knowledge

information in a systematic way, which can also guarantee stability and

performance criteria.

The SMC method is proposed to provide robust con-

troller systems,

1011

However, this approach needs a nominal mathematical

model for the control law. Fuzzy control technique has been successfully ap-

plied to many industrial systems where no accurate mathematical models of

the systems under control are available and human experts are available to

provide linguistic fuzzy control rules or linguistic fuzzy descriptions about

the systems,

The apparent similarities between sliding mode control and

fuzzy control motivate considerable research eﬀorts in combining the two

approaches for achieving more superior performances such as overcoming

some limitations of the traditional sliding mode control,

In this paper,

adaptive fuzzy controller is proposed for a class of nonlinear systems with

unknown nonlinear dynamics. The adaptive fuzzy model type T-S is used

to approximate the unknown dynamics systems. The adjustable fuzzy pa-

rameters are updated on line by the adaptive algorithm. The stability and

convergence analysis is ensured from the Lyapunov approach. The added

term control based on SMC approach is designed to attenuate the inﬂuence

of external disturbances and to remove the fuzzy approximation error. This

paper is organized as follows: The problem formulation is presented in sec-

tion 2. In section 3, the fuzzy SMC design is proposed for nonlinear systems

with unknown dynamics. In Section 4, simulations example is proposed to

shown the eﬀectiveness of the proposed method.

2. Problem formulation

Consider a nonlinear system described by:











˙x

= x

˙x

= x

˙x

= f(x

, t) + bu + d(t)

y = x

(1)

or equivalently



(n)

= f(x

, t) + bu + d(t)

y = x

with x

(n)

(2)

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357

x = [x, ˙x, . . . , x

(n−1)

]

= [x

, x

, . . . , x

]

∈ IR

is the state vector of the

systems which is assumed to be available for measurements, u ∈ IR and y ∈

IR are respectively the input and the output of the system. f(x, t) is un-

known and nonlinear function, and b is positive constant. d(t) is unknown

external disturbances. The control problem is to obtain the state x for

tracking a desired y

. Deﬁne the tracking error e = y − y

. It is desired

that the output error of the system follow: e

(n)

+ k

n−1

(n−1)

+ ···+ k

e = 0

to ensure a good tracking. With k

, i = 1, 2, . . . , n −1 are selected so that

the associated characteristic equation has roots in the open left-half com-

plex. When the system (1) is well known without disturbances and b 6= 0,

the control law is designed to guarantee that lim e

t→∞

= 0 as follows:

u =

(−f(x, t) + y

(n)

−

n−1

i=1

(i)

). (3)

However, in practice, this objective can not be achieved the dynamic of sys-

tem f(x

, t) is unknown and the design of the control law (3) is confronted

with many problems due to environment changes, modelling errors and un-

modelled dynamics. To solve this problem, we propose a fuzzy system to

approximate the unknown dynamic and using the SMC technique and Lya-

punov approach to guarantee the stability and the tracking performance.

3. Fuzzy sliding mode control design

In this section, the adaptive fuzzy control is designed for a class of nonlinear

system with unknown nonlinear dynamics and disturbances. The strategy

of control is based on fuzzy logic and SMC approach to ensure stability

and good tracking. The dynamics system f(x

, t) is approximated by the

adaptive fuzzy model type T-S. The fuzzy parameters can be tuned on

line by adaptive law based on Lyapunov approach. To ensure stability and

tracking performance, the auxiliary control action is incorporated in the

control law (3) to attenuate the external disturbances and remove fuzzy

approximation error. The design of this compensation control is derived

from SMC and the Lyapunov approach. The basic conﬁguration of a fuzzy

logic system consists of a fuzziﬁer, a fuzzy rule base, a fuzzy inference engine

and a defuzziﬁer. The fuzzy inference engine uses the fuzzy IF-THEN rules

to perform a mapping from an input vector x = [x

, x

, . . . , x

]

to an

output scalar

f. The fuzzy rule base consists of a collection of fuzzy IF-

THEN rules in the following form:

: if x

is F

and . . . and x

is F

, then

f is θ

j = 1, . . . , M (4)

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358

where F

, i = 1, . . . , n are fuzzy variables characterized by membership

functions µ

) and θ

is the corresponding value of the output fuzzy

singleton.

The output of the fuzzy system with singleton fuzziﬁcation, product infer-

ence and center average defuzziﬁcation can be expressed as:

f(x

, θ) =

j=1

(

i=1

) )

j=1

(

i=1

) )

= θ

ξ(x

) (5)

M is the total number of the fuzzy rules, θ = [θ

, θ

, . . . , θ

]

is adjustable

parameter vector,ξ(x

) = [ξ

(x), ξ

(x) , . . . , ξ

(x) ]

is the vector of the

fuzzy basis functions [7]:

(x) = (

i=1

) )



j=1

(

i=1

) ) (6)

Deﬁne the optimal parameters vectors and fuzzy approximation error as:

∗

= arg min

θ∈Ω



sup

x∈IR



f(x

, t) −

f(x, θ)





(7)

Ω

= {θ

∈ IR

kθk ≤ M

} is the convex compact sets which contain feasi-

ble parameter sets for θ , with M

is given constants. Deﬁne a time varying

sliding surface s(x, t) = 0 , in the state space IR

by the scalar equation:

s(x, t) = e

(n−1)

+ k

n−1

(n−2)

+ ···+ k

(8)

Assumptions:

i) There exist a positive constant f

max

such that



f(x

, t) −

f(x, θ)



max

ii) The positive constant k

is chosen so that the following condition: k

≥

max

+ d

max

is satisﬁed with d

max

is a positive upper bound of the

external disturbances.

The proposed control law is given by:

u = ˆu(x

, θ) + u

(9)

The primary control is represented as follows:

ˆu(x, θ) =

(−

f(x

, θ) + y

(n)

−

n−1

i=1

(i)

) (10)

The auxiliary control is given as:

= −k

tanh





(11)

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359

with tanh





is the hyperbolic tangent of

and ε is a small positive con-

stant. The adjustable fuzzy parameters of

f(x

, θ) are tuned on line using

the Lyapunov approach. In order to guarantee that the parameters are

bounded, we introduce the projection algorithm [9] to restrict them in the

closed set Ω .

θ =







−γsξ(x) if (|θ| < M or (|θ| = M and γsξ

ξ(x) > 0))

−γsξ(x) + γs

θθ

ξ(x)

|θ|

(12)

γ is a positive constant.

Theorem 3.1. Consider the nonlinear system (1), satisfying the assump-

tions i) and ii). The fuzzy SMC law is chosen as (9) with The closed-loop

system is stable in sense that all the signals are bounded and the tracking

performance is achieved.

Proof:

Consider the following Lyapunov function:

V =

2γ

(13)

With Φ = θ − θ

∗

. The time derivative of V equals:

V = s ˙s +

= s((f (x, t) −

f(θ, x)) + d(t) + u

) +

= s((f (x, t) −

∗

(θx)) + (

∗

(θ, x) − f(θ, x)) + d(t) + u

) +

= s(Φ

ξ(x) + (

∗

(θ, x) − f(x, t)) + d(t) + u

) +

(

+ γ sξ(x)) + s((

∗

(θ, x) − f(x, t)) + d(t) + u

)

V ≤

(

Φ + γ sξ(x)) + |s|(f

max

+ d

max

) + s u

(14)

where

Φ =

θ ;

V ≤

(

θ + γ sξ(x)) + |s|(f

max

+ d

max

) + s u

The adjustable fuzzy parameters of

f(x, θ) is chosen as follows:

θ =

−γ sξ(x) Then we have:

V ≤ |s|(f

max

+ d

max

) + s u

(15)

if |s| > ε, tanh (

) = sgn(s) , then u

= −k

sgn(s) we obtain

V ≤

|s| (f

max

+ d

max

) − k

|s| ,

By choosing the positive constant k

≥ f

max

+ d

max

we obtain

V ≤ 0 .